New experimental techniques give a far more detailed picture of the motion of cellular components than was previously available. In single-particle tracking (SPT), computer-enhanced video microscopy is used to measure the 2d trajectories of labeled membrane proteins or lipids on the cell surface, and the 3d trajectories of proteins, nucleic acids, and various subcellular structures in the cytoplasm and nucleus. Typically the spatial resolution is tens of nanometers and the time resolution is tens of milliseconds. One of the major results of SPT is that a significant fraction of proteins and lipids in the plasma membrane undergo various types of non-Brownian motion, including anomalous subdiffusion, directed motion, and confined motion. Transitions are often observed between modes of motion. A similar picture is emerging for 3d motion within the cell. This project will use Monte Carlo computer simulation techniques and percolation theory to study heterogeneous motion in heterogeneous membranes, and its biological consequences. The work will examine various models of hindered diffusion, such as the effect of the screened electrostatic interaction of highly charged extracellular domains of membrane proteins. The effects of membrane heterogeneity on reaction kinetics will be studied, specifically the role of lipid rafts in cellular signal transduction. Simulations will be used to develop improved methods of SPT analysis for both the 2d and 3d cases. Other 3d work will model diffusion obstructed by the simplest geometric model of the cytoskeleton, and SPT of Cajal bodies in the nucleus.